Octahedron - Area and Volume

Area and Volume

The surface area A and the volume V of a regular octahedron of edge length a are:

Thus the volume is four times that of a regular tetrahedron with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).

If an octahedron has been stretched so that it obeys the equation:

The formula for the surface area and volume expand to become:

Additionally the inertia tensor of the stretched octahedron is:

I = \begin{bmatrix} \frac{1}{10} m (y_m^2+z_m^2) & 0 & 0 \\ 0 & \frac{1}{10} m (x_m^2+z_m^2) & 0 \\ 0 & 0 & \frac{1}{10} m (x_m^2+y_m^2) \end{bmatrix}

These reduce to the equations for the regular octahedron when:

Read more about this topic:  Octahedron

Famous quotes containing the words area and/or volume:

    “Many women are reluctant to allow men to enter their domain. They don’t want men to acquire skills in what has traditionally been their area of competence and one of their main sources of self-esteem. So while they complain about the male’s unwillingness to share in domestic duties, they continually push the male out when he moves too confidently into what has previously been their exclusive world.”
    —Bettina Arndt (20th century)

    “Love is both Creator’s and Saviour’s gospel to mankind; a volume bound in rose-leaves, clasped with violets, and by the beaks of humming-birds printed with peach-juice on the leaves of lilies.”
    —Herman Melville (1819–1891)